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In mathematics, a '''Grothendieck category''' is a certain kind of [[abelian category]], introduced by [[Alexander Grothendieck]] in 1957<ref>{{citation|first=A.|last=Grothendieck|authorlink=Alexander Grothendieck|title=Sur quelques points d’algèbre homologique|journal=[[Tôhoku Mathematical Journal]]|volume=9|series=(2)|pages=119–221|year=1957|mr=0102537|url=http://projecteuclid.org/euclid.tmj/1178244839}}. [ftp://ftp.math.mcgill.ca/barr/pdffiles/gk.pdf English translation].</ref> in order to develop the machinery of [[homological algebra]] for [[module (mathematics)|module]]s and for [[sheaf (mathematics)|sheaves]] in a unified manner.
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To every [[algebraic variety]] ''V'' one can associate a Grothendieck category qcoh(''V''), consisting of the [[quasi-coherent sheaf|quasi-coherent sheaves]] on ''V''. This category encodes all the relevant information about ''V'', and ''V'' can be recovered from qcoh(''V''). This example gives rise to one approach to [[noncommutative algebraic geometry]]: the study of "non-commutative varieties" is then nothing but the study of Grothendieck categories.<ref>{{cite web|title=Quantum Ruled Surfaces|author=Izuru Mori|year=2007|url=http://mathsoc.jp/section/algebra/algsymp_past/algsymp07_files/mouri.pdf}}</ref>
 
==Definition==
By definition, a Grothendieck category ''A'' is an [[AB5 category]] with a [[Generator (category theory)|generator]]. Spelled out, this means that
* ''A'' is an [[abelian category]];
* every (possibly infinite) family of objects in ''A'' has a [[coproduct]] (a.k.a. direct sum) in ''A'';
* [[direct limit]]s (a.k.a. filtered colimits) of [[exact sequence]]s are exact; this means that if a direct system of [[short exact sequence]]s in ''A'' is given, then the induced sequence of direct limits is a short exact sequence as well. (Direct limits are always [[exact functor|right-exact]]; the important point here is that we require them to be [[exact functor|left-exact]] as well.)
* ''A'' possesses a generator, i.e. there is an object ''G'' in ''A'' such that Hom(''G'',&nbsp;–) is a [[faithful functor]] from ''A'' to the [[category of sets]]. (In our situation, this is equivalent to saying that every object ''X'' of ''A'' admits an [[epimorphism]] ''G''<sup>(''I'')</sup>→''X'', where ''G''<sup>(''I'')</sup> denotes a direct sum of copies of ''G'', one for each element of the (possibly infinite) set ''I''.)
 
==Examples==
* The prototypical example of a Grothendieck category is the [[category of abelian groups]]; the group '''Z''' of integers can serve as a generator.
* More generally, given any [[ring (mathematics)|ring]] ''R'' (associative, with 1, but not necessarily commutative), the category Mod-''R'' of all left (or alternatively: right) [[module (mathematics)|module]]s over ''R'' is a Grothendieck category; ''R'' itself can serve as a generator.
* Given a [[topological space]] ''X'', the category of all [[sheaf (mathematics)|sheaves]] of abelian groups on ''X'' is a Grothendieck category. (More generally: the category of all sheaves of left ''R''-modules on ''X'' is a Grothendieck category for any ring ''R''.
* Given a [[ringed space]] (''X'',''O<sub>X</sub>''), the category of [[Sheaf_(mathematics)#Sheaves_of_modules|sheaves of ''O<sub>X</sub>''-modules]] is a Grothendieck category.
* Any category that's [[equivalence of categories|equivalent]] to a Grothendieck category is itself a Grothendieck category.
* Given a [[Category_(mathematics)#Small_and_large_categories|small category]] ''C'' and a Grothendieck category ''A'', the [[functor category]] Funct(''C'',''A'') is a Grothendieck category; if ''C'' is [[preadditive category|preadditive]], then the functor category Add(''C'',''A'') of all additive functors from ''C'' to ''A'' is a Grothendieck category as well.
* Given an (affine or projective) [[algebraic variety]] ''V'' (or more generally: a [[quasi-compact]] [[Glossary_of_scheme_theory#Separated_and_proper_morphisms|quasi-separated]] [[scheme (mathematics)|scheme]]), the category of [[quasi-coherent sheaf|quasi-coherent sheaves]] on ''V'' is a Grothendieck category.
 
==Properties==
Every object in a Grothendieck category ''A'' has an [[injective hull]] in ''A''. This allows to construct [[injective resolution]]s and thereby the use of the tools of [[homological algebra]] in ''A''. Further, ''A'' contains an [[injective cogenerator]].
 
In a Grothendieck category, any family of [[subobject]]s (''U''<sub>''i''</sub>) of a given object ''X'' has a [[supremum]] Σ<sub>''i''</sub>''U''<sub>''i''</sub> which is again a subobject of ''X''. Further, if the family (''U''<sub>''i''</sub>) is directed (i.e. for any two objects in the family, the family has a third object that contains the two), and ''V'' is another subobject of ''X'', we have
:<math>\sum_{i}(U_i\cap V) = \left(\sum_{i}U_i\right) \cap V.</math>
 
In a Grothendieck category, arbitrary [[limit (category theory)|limit]]s (and in particular [[product (category theory)|product]]s) exist. It follows directly from the definition that arbitrary [[limit (category theory)|colimits]] and [[coproduct]]s (direct sums) exist as well. We can thus say that every Grothendieck category is [[complete category|complete and co-complete]]. Coproducts in a Grothendieck category are exact (i.e. the coproduct of a family of short exact sequences is again a short exact sequence), but products need not be exact.
 
If ''A'' is a Grothendieck category and ''T'' is a [[localizing subcategory]] of ''A'' (i.e. ''T'' is a full subcategory of ''A'', closed under ''A''-colimits, and for any [[short exact sequence]] <math>0\rarr A_1\rarr A_2\rarr A_3\rarr 0</math> in ''A'', ''A<sub>2</sub>'' is in ''T'' if and only if ''A<sub>1</sub>'' and ''A<sub>3</sub>'' are in ''T''), we can form the [[Serre quotient category]] ''A''/''T''. This quotient is again a Grothendieck category.
 
The [[Gabriel–Popesco theorem]] states that any Grothendieck category ''A'' is equivalent to a [[full subcategory]] of the category Mod-''R'' of right modules over some unital ring ''R'' (which can be taken to be the [[endomorphism ring]] of a generator of ''A''), and ''A'' can be obtained as a Serre quotient of Mod-''R'' by some localizing subcategory.<ref>{{cite journal |authors=N. Popesco, P. Gabriel|title=Caractérisation des catégories abéliennes avec générateurs et limites inductives exactes|journal=Comptes Rendus de l'Académie des Sciences|volume=258|year=1964|pages=4188–4190}}</ref>
 
==References==
{{reflist}}
*{{cite book|author=N. Popescu|title=Abelian categories with applications to rings and modules|publisher=Academic Press|year=1973}}
*{{citation|first1=Pascual|last1=Jara|first2=Alain|last2=Verschoren|first3=Conchi|last3=Vidal|title=Localization and sheaves: a relative point of view|series=Pitman Research Notes in Mathematics Series|volume=339|publisher=Longman, Harlow|year=1995}}.
 
==External links==
*{{SpringerEOM
  | title=Grothendieck category
| id=Grothendieck_category
| last=Tsalenko
| first=M.Sh.  
  | author-link=
| last2=
| first2=
| author2-link=
}}
*[http://therisingsea.org/notes/AbelianCategories.pdf Abelian Categories], notes by Daniel Murfet. Section 2.3 covers Grothendieck categories.
 
[[Category:Homological algebra]]
[[Category:Category theory]]

Latest revision as of 18:42, 23 October 2014

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This is crucial for the reason that a full tang knife gives great grip even even though its handle gets broken. Paring knife also comes with plastic deal with but most of the time it is fantastic to stay clear of such knives that comes with plastic handle. Most of paring knife comes with 3 to four inch blade which is most of the time made up of high stainless steel carbon. Paring knife can be categories in few distinctive sort which is known as ceramic knives, farberware knives, shenzhen knives, victronix knives and so forth.

First of all it can get pretty sharp, but that is about the only factor it has going for it. Sharpening it is a thing that has to be performed by a specialist, it really is also a lot more fragile than a steel knife. Use disposable things like aprons, paper chef hats, hair nets, to-go containers, paper napkins, coffee stirs and straws, and a lot more for providing prospects with carry out merchandise or catering service.

I have recently been receiving additional and much more into cooking and this report reminds me I need to have a superior Chef's knife. My flat has about twenty knives - and they have been all blunt when I moved in (I believe they just bought more as the old ones blunted...) so i purchased a knife sharpener, and we most likely won't need to have a new knife for a lengthy time. I took cooking class before in High college and the only knives we applied were the chef's knife and the paring knife.

This 14-piece set stocks a kitchen with most cutlery wants, and packs all knives in a modern black hardwood storage block. The set includes: an eight-inch chef's knife, an eight-inch slicer, an 8-inch bread knife, a 5-inch boning knife, a 4-1/2-inch utility knife, a 3-1/2-inch paring knife, and six four-1/two-inch steak knives, plus a pair of shears. The second season of Best Chef started airing in October 2006.

As I said just before a Chef's knife is an all-porpoise knife and it can be made use of for nearly any operation that regards cutting. In reality as a main item, in your kitchen you won`t need to have any other type of knife that requires slicing, chopping, mincing and so on. At initially here it is one particular tip : First peel the outer brown layer with the help of the knife and fingers. Steak knife - is an optional knife to the dining table.

Lots of of us can name a couple of knife brands but not all of them are identified for their top quality. Just before you buy a knife or knife set be positive to genuine read consumer testimonials and not just a single critique on a site. You want to see what more than just a handful of people today have to say so that you can get a really superior concept of how your knife will perform and hold up. That is why you need to have to stick with extremely visible brands with a lot of buyer evaluations. This is the well-liked French knife and German knife.

Dojosaki Bocho- This knife has been specially made for filleting fishes, especially the sea eel and other tiny fishes that are caught in the course of the summer time. Santoku knife- This Japanese chef knife comes with a scalloped edge and is ideally suited for scalloping many types of meats and fishes. The popular length of the chef knife is eight to 12 inches. Most chef handles are made from wood, but wood does have specific issues. I got my wife a Wusthof set for Christmas.

They are all good (at least the brand ones), but when you do come across the chef knife that is finest suited for you, you will know. A really sharp chef knife is a single that there is no need to have to apply force when you reduce with it. When you talk about French knife, you will find out that this knife has really long history in cooking schools. This kind of knife is popularly use for cooking schools since of its extended curve at the tip of the blade.